Summary: Recently it has been argued that the ground state of high density QCD is likely be a combination of the CFL-phase along with condensation of the \(K^{0}\) field. This state spontaneously breaks a global \(U(1)_{Y}\) symmetry, therefore one would expect the formation of \(U(1)_{Y}\) global strings. We discuss the core structure of these strings and demonstrate that under some conditions the global \(U(1)_{Y}\) symmetry may not be restored inside the string, in contrast with the standard expectations. Instead, \(K^{+}\) condensation occurs inside the core of the string if a relevant parameter cos \(\theta _{K^{0}}\equiv m_{K^{0}}^{2}/\mu_{eff^{2}}\) is larger than some critical value \(\theta _{K^{0}} \geq \theta_{crit}\). If this phenomenon happens, the \(U(1)_{Y}\) strings become superconducting and may considerably influence the magnetic properties of dense quark matter, in particular in neutron stars.